The open smooth case

Let be a smooth variety and a smooth compactification, such that is a (smooth) normal crossing divisor. Consider the long exact sequence of relative cohomology groups

By excision and by Thom isomorphism, we have
where is the normal bundle of in . Thus, we have that can be seen as a pure Hodge structure but of the wrong weight, since we would like
to be a morphism of pure Hodge structures (of bidegree ) but they have weights and . Thus, we take the Tate twist so that our long exact sequence looks like
admits a mixed Hodge structure with induced weight filtration
Where does the decreasing filtration arise? In the classical case, i.e. for a complex manifold , we consider the complex of sheaves on of holomorphic -forms and its naive filtration
It is easy to check that induces a decreasing filtration on the hypercohomology of , which is defined as
Since is a resolution of the locally constant sheaf of stalk over , we have
which induces a decreasing filtration on . In the case of compact Kähler manifolds, this is what we are looking for and, indeed, the filtration leads us to the proof of (a weaker version of) the Hodge decomposition. Unfortunately, if we try to use the same construction for , in general we get no extra information. For instance, for affine,we would get
We need to replace our complex of sheaves with something clever. It turns out that if we allow logarithmic singularities along , we are able to induce a (non-trivial) decreasing filtration on from the naive one on the new complex. More precisely, let be the subcomplex of , where is the inclusion of in its compactification such that

  • a meromorphic differential -form on an open , holomorphic on is an element of if and admit poles of order at most 1 in .

We can give an explicit local description of the elements of : Let be local coordinates on an open set of , in which is defined by the equation

Then, the elements of the form
for , , , form a basis of .

Theorem 11.2

 

Hence, we can consider the filtration induced by the naive

Such a filtration induces a pure Hodge structure of weight on

Remark 11.1

In the case of singular, we take a smooth resolution with exceptional set , where are simple normal crossing divisors. We iterate the procedure briefly explained in the previous section in a similar manner. Moreover, if is proper and surjective, then

 
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